You can manage inquiries in non-business hours with ticket.
For the setting of business hours, please refer to Setting of Chat reception hours (business hours).
When receiving inquiries with emails, it is hard to see if the inquiries have been solved or not, however, with ticket, it would manage inquiries by sharing as task method, so you would not miss unsolved inquiries.
For more information about setting to use ticket, please refer to use ticket function.